Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #491760 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | D4×D8 |
| commutator, Frattini, cosocle | C4×C8 | C24 |
| prime radical | #19968 | C2 |
| LCS | G ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C42 ◁ C8×D8 ◁ G | |
| LECS | G ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
476 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 40 points:
(2,5)(3,8)(6,7)(9,10,12,15)(11,19,20,37)(13,25,23,39)(14,28,24,18)(16,32,30,21)(17,34,31,27)(22,38,29,35)(26,36,40,33);(1,2)(3,6)(4,7)(5,8)(10,15)(16,30)(17,31)(18,28)(19,37)(25,39)(33,36)(35,38);(9,11)(10,16)(12,20)(13,26)(14,29)(15,30)(17,35)(18,36)(19,25)(21,27)(22,24)(23,40)(28,33)(31,38)(32,34)(37,39);(2,6)(3,8)(5,7)(11,20)(16,30)(19,37)(21,32)(22,29)(26,40)(33,36)(35,38)Polycyclic presentation:
2^9|a^2=i,e^2=g,f^2=hi,h^2=i,b^a=be,c^a=cf,e^a=eg,f^a=fh,h^a=hi,d^b=dei,e^b=eg,d^c=di,f^c=fh,h^c=hi,e^d=eg50 conjugacy classes — view the full character table.
| maximal subgroup of | #72310003 |
| maximal normal subgroup of | #72310003 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 398964);